DEGENERA - Colorado State Universitymiranda/preprints/rigateZappa2CCFM.pdfZappa's argumen ts rely on a rather in tricate analysis concerning degenerations of h yp er-plane sections

DEGENERATIONS OF SCROLLS TO UNIONS OF PLANESA. CALABRI, C. CILIBERTO, F. FLAMINI, R. MIRANDAAbstract. In this paper we study degenerations of scrolls to union of planes, a problemalready considered by G. Zappa in [25] and [26]. We prove, using techniques di�erent from theones of Zappa, a degeneration result to union of planes with the mildest possible singularities,for linearly normal scrolls of genus g and of degree d � 2g + 4 in Pd�2g+1. We also studyproperties of components of the Hilbert scheme parametrizing scrolls. Finally we reviewZappa's original approach.Dedicated to Professor G. Zappa on his 90th birthday1. IntroductionIn this paper we deal with the problem, originally studied by Guido Zappa in [25, 26],concerning the embedded degenerations of two-dimensional scrolls, to union of planes withthe simplest possible singularities.In [2] and [3], we have studied the properties of the so-called Zappatic surfaces, i.e. re-duced, connected, projective surfaces which are unions of smooth surfaces with global normalcrossings except at singular points, which are locally analytically isomorphic to the vertex ofa cone over a union of lines whose dual graph is either a chain of length n, or a fork with n�1teeth, or a cycle of order n, and with maximal embedding dimension. These singular pointsare respectively called (good) Zappatic singularities of type Rn, Sn and En (cf. De�nition 2.1below). A Zappatic surface is said to be planar if it is embedded in a projective space andall its irreducible components are planes.An interesting problem is to �nd degenerations of surfaces to Zappatic surfaces with Zap-patic singularities as simple as possible. This problem has been partly considered in [3]; e.g.in Corollary 8.10, it has been shown that, if X is a Zappatic surface which is the at limit ofa smooth scroll of sectional genus g � 2, then the Zappatic singularities of X cannot be toosimple, in particular X has to have some point of type Ri or Si, with i � 4, or of type Ej,with j � 6.The main results in [25] can be stated in the following way:Theorem 1.1. (cf. x12 in [25]) Let F be a scroll of sectional genus g, degree d � 3g+2, whosegeneral hyperplane section is a general curve of genus g. Then F is birationally equivalentto a scroll in Pr, for some r � 3, which degenerates to a planar Zappatic surface with onlypoints of type R3 and S4 as Zappatic singularities.Zappa's arguments rely on a rather intricate analysis concerning degenerations of hyper-plane sections of the scroll and, accordingly, of the branch curve of a general projection ofthe scroll to a plane.We have not been able to check all the details of this very clever argument. However, wehave been able to prove a slightly more general result using some basic smoothing technique(cf. [6]).Mathematics Subject Classi�cation (2000): 14J26, 14D06, 14N20; (Secondary) 14H60, 14N10.The �rst three authors are members of G.N.S.A.G.A. at I.N.d.A.M. \Francesco Severi".1

2 A. CALABRI, C. CILIBERTO, F. FLAMINI, R. MIRANDAOur main result is the following (cf. Proposition 3.8, Constructions 4.1, 4.2, Remarks 4.20,5.6 and Theorems 4.6, 5.4 later on):Theorem 1.2. Let g � 0 and either d � 2, if g = 0, or d � 5, if g = 1, or d � 2g + 4, ifg � 2. Then there exists a unique irreducible component Hd;g of the Hilbert scheme of scrollsof degree d and sectional genus g in Pd�2g+1, such that the general point of Hd;g represents asmooth scroll S which is linearly normal, i.e. H1(S;OS(1)) = 0.Furthermore,(i) Hd;g is generically reduced and dim(Hd;g) = (d� 2g + 2)2 + 7(g � 1),(ii) Hd;g contains the Hilbert point of a planar Zappatic surface having only d � 2g + 2points of type R3 and 2g � 2 points of type S4 as Zappatic singularities,(iii) Hd;g dominates the moduli space Mg of smooth curves of genus g.We also construct examples of scrolls S with same numerical invariants, which are notlinearly normal in Pd�2g+1, as well as examples of components of the Hilbert scheme of scrollswith same invariants, di�erent from Hd;g and with general moduli (cf. Examples 5.11 and5.12).We shortly describe the contents of the paper. In x 2 we recall standard de�nitions andproperties of Zappatic surfaces. In x 3 we focus on some degenerations of products of curvesto planar Zappatic surfaces and we prove some results which go back to [26]. In particular,we consider Zappatic degeration of rational and elliptic normal scrolls and of abelian surfaces.In x 4 we prove the greatest part of Theorem 1.2. First, we construct, with an inductiveargument, planar Zappatic surfaces which have the same numerical invariants of scrolls ofdegree d and genus g in Pd�2g+1 and having only d � 2g + 2 points of type R3 and 2g � 2points of type S4 as Zappatic singularities. Then we prove that these Zappatic surfaces can besmoothed to smooth scrolls which �ll up the componentHd;g and we compute the cohomologyof the hyperplane bundle and of the normal bundle. These computations imply that Hd;g isgenerically smooth, of the right dimension and its general point represents a linearly normalscroll.Section 5 is devoted to study some properties of components of the Hilbert scheme ofscrolls. In particular, we show that the component Hd;g is the unique component of theHilbert scheme of scrolls of degree d and sectional genus g whose general point is linearlynormal in Pd�2g+1. Moreover we give the examples mentioned above (cf. Examples 5.11 and5.12).In the last section, x 6, we brie y explain Zappa's original approach in [25]. Moreover,we make some comments and give some improvements on some interesting results from [25]concerning extendability of plane curves to scrolls which are not cones.2. Notation and preliminariesIn this paper we deal with projective varieties de�ned over the complex �eld C .Let us recall the notions of Zappatic singularities, Zappatic surfaces and their dual graphs.We refer the reader for more details to our previous papers [2] and [3]. One word of warning:what we call good Zappatic singularities there, here we simply call Zappatic singularities,because no other type of Zappatic singularity will be considered in this paper.De�nition 2.1. Let us denote by Rn [resp. Sn, En] a graph which is a chain [resp. a fork,a cycle] with n vertices, n � 3, cf. Figure 1. Let CRn [resp. CSn, CEn] be a connected,projectively normal curve of degree n in Pn [resp. in Pn, in Pn�1], which is a stick curve, i.e.a reduced, union of lines with only double points, whose dual graph is Rn [resp. Sn, En].

DEGENERATIONS OF SCROLLS TO UNIONS OF PLANES 3� � � � � � � �� � � ��� ���� � � �Figure 1. A chain Rn, a fork Sn with n� 1 teeth, a cycle En.We say that a point x of a projective surface X is a point of type Rn [resp. Sn, En] if(X; x) is locally analytically isomorphic to a pair (Y; y) where Y is the cone over a curve CRn[resp. CSn, CEn], n � 3, and y is the vertex of the cone (cf. Figure 2). We say that Rn-, Sn-,En-points are Zappatic singularities.�V1 V2 V3C12 C23

�V1 V2 V3C12 C23V4C24

�V1 V2V3C13 C12 C23Figure 2. Examples: a R3-point, a S4-point and an E3-point.In this paper we will deal mainly with points of type R3 and S4. We will use the following:Notation 2.2. If x is a point of type R3 [of type S4, resp.] of a projective surface X, we saythat the component V2 of X as in picture on the left [in the middle, resp.] in Figure 2 is thecentral component of X passing through x.De�nition 2.3. A projective surface X = Svi=1 Vi is called a Zappatic surface if X is con-nected, reduced, all its irreducible components V1; : : : ; Vv are smooth and:� the singularities in codimension one of X are at most double curves which are smoothand irreducible along which two surfaces meet transversally;� the further singularities of X are Zappatic singularities.We set Cij = Vi \ Vj if Vi and Vj meet along a curve, we set Cij = ; otherwise. We setCi = Vi \ X � Vi = Svj=1Cij. We denote by C = Sing(X) the singular locus of X, i.e. thecurve C = S1�i<j�v Cij.We denote by fn [resp. rn, sn] the number of point of type En [resp. Rn, Sn] of X.Remark 2.4. A Zappatic surface X is Cohen-Macaulay. Moreover it has global normalcrossings except at the Rn- and Sn-points, for n � 3, and at the Em-points, for m � 4.We associate to a Zappatic surface X a dual graph GX as follows.De�nition 2.5. Let X = Svi=1 Vi be a Zappatic surface. The dual graph GX of X is givenby: � a vertex vi for each irreducible component Vi of X;� an edge lij, joining the vertices vi and vj, for each irreducible component of the curveCij = Vi \ Vj;

4 A. CALABRI, C. CILIBERTO, F. FLAMINI, R. MIRANDA� a n-face Fp for each point p of X of type En for some n � 3: the n edges boundingthe face Fp are the n irreducible components of the double curve C of X concurringat p;� an open n-face for each point p of X of type Rn for some n � 3; it is bounded byn � 1 edges, corresponding to the n � 1 irreducible components of the double curveof X concurring at p, and by a dashed edge, which we add in order to join the twoextremal vertices;� a n-angle for each p of X of type Sn, spanned by the n� 1 edges that are the n� 1irreducible components of the double curves of X concurring at p.By abusing notation, we will denote by GX also the CW-complex associated to the dual graphGX of X, formed by vertices, edges and n-faces.Remark 2.6 (cf. [2]). When we deal with the dual graph of a planar Zappatic surfaceX = Svi=1 Vi, we will not indicate open 3-faces with a dashed edge. Indeed, the graph itselfshows where open 3-faces are located.Some invariants of a Zappatic surface X have been computed in [2] and in [4], namelythe Euler-Poincar�e characteristic �(OX), the !-genus p!(X) = h0(X;!X), where !X is thedualizing sheaf of X, and, when X is embedded in a projective space Pr, the sectional genusg(X), i.e. the arithmetic genus of a general hyperplane section of X. In particular, for aplanar Zappatic surface (for the general case, see [2, 4]) one has:Proposition 2.7. Let X = Svi=1 Vi be a planar Zappatic surface of degree v in Pr and denoteby e the degree of C = Sing(X), i.e. the number of double lines of X. Then:g(X) = e� v + 1; (2.8)p!(X) = h0(X;!X) = h2(GX ; C ); (2.9)�(OX) = �(GX) = v � e +Xi�3 fi: (2.10)In this paper, a Zappatic surface will always be considered as the central �bre of an em-bedded degeneration, in the following sense.De�nition 2.11. Let � be the spectrum of a DVR (or equivalently the complex unit disk).A degeneration of surfaces parametrized by � is a proper and at morphism � : X ! � suchthat each �bre Xt = ��1(t), t 6= 0 (where 0 is the closed point of �), is a smooth, irreducible,projective surface. A degeneration � : X ! � is said to be embedded in Pr if X � � � Prand the following diagram commutes: X� � �� Prpr1�The invariants of the Zappatic surface X = X0, which is the central �bre of an embeddeddegeneration X ! �, determine the invariants of the general �bre Xt, t 6= 0, as we proved in[2, 3, 4]. Again, we recall these results only for planar Zappatic surfaces and we refer to ourprevious papers for the general case.Theorem 2.12. Let X ! � be an embedded degeneration in Pr such that the central �breX = X0 is a planar Zappatic surface. Then, for any 0 6= t 2 �:g(Xt) = g(X); pg(Xt) = p!(X); �(OXt) = �(OX): (2.13)

DEGENERATIONS OF SCROLLS TO UNIONS OF PLANES 5Moreover the self-intersection K2Xt of a canonical divisor of Xt is:K2Xt = 9v � 10e+Xn�3 2nfn + r3 + k; (2.14)where k depends on the presence of points of type Rm and Sm, m � 4:Xm�4(m� 2)(rm + sm) � k �Xm�4(2m� 5)rm + �m� 12 �sm:Finally, let us recall the construction of rational normal scrolls.De�nition 2.15. Fix two positive integers a; b and set r = a+b+1. In Pr choose two disjointlinear spaces Pa and Pb. Let Ca [resp. Cb] be a smooth, rational normal curve of degree a inPa [resp. of degree b in Pb] and �x an isomorphism � : Ca ! Cb. Then, the union in Pr ofall the lines p; �(p), p 2 Ca, is a smooth, rational, projectively normal surface which is calledscroll of type (a; b) and it is denoted by Sa;b. Such a scroll is said to be balanced if eitherb = a or b = a+ 1.Another way to de�ne a scroll is as the embedding of a Hirzebruch surface Fn , n � 0,which is the minimal ruled surface over P1 with a section of self-intersection (�n). Setting Fthe ruling of Fn and C a section such that C2 = n, the linear system jC + aF j embeds Fn inPn+2a+1 as a scroll of type (a; a+n), cf. e.g. [14]. In particular a balanced scroll in Pr, r � 3,is the embedding either of F0 = P1 � P1 or of F1 depending on whether r is odd or even.In the next section we will see, in particular, degenerations of rational scrolls to a planarZappatic surface. In the subsequent section we will deal with scrolls of higher genus.3. Degenerations of product of curves and of rational scrollsZappa suggested in [26] an interesting method for degenerating products of curves, whichalso gives a degeneration of rational and elliptic scrolls to planar Zappatic surfaces with onlyR3-points.Example 3.1 (Zappa). Let C � Pn�1 and C 0 � Pm�1 be smooth curves. If C and C 0 maydegenerate to stick curves, then the smooth surfaceS = C � C 0 � Pn�1 � Pm�1 � Pnm�1;embedded via the Segre map, degenerates to a Zappatic surface Y in Pnm�1 whose irreduciblecomponents are quadrics and whose double curves are lines.If it is possible to further, independently, degenerate each quadric of Y to the union oftwo planes, then one gets a degeneration of S = C � C 0 to a planar Zappatic surface. Thiscertainly happens if each quadric of Y meets the other quadrics of Y along a union of at mostfour lines, at most two from each ruling (see Figure 3).Therefore S = C � C 0 can degenerate to a planar Zappatic surface if C and C 0 are ei-ther rational or elliptic normal curves, since they degenerate to stick curves CRn and CEn,respectively. We will now describe these degenerations.Example 3.2 (Rational scrolls). Let C be a smooth, rational normal curve of degree n in Pn.Since C degenerates to a union of n lines whose dual graph is a chain, the smooth rationalnormal scroll S = C � P1 � P2n+1 degenerates to a Zappatic surface Y = Sni=1 Yi such thateach Yi is a quadric, Y has no Zappatic singularity and its dual graph GY is a chain of lengthn, see Figure 4.

6 A. CALABRI, C. CILIBERTO, F. FLAMINI, R. MIRANDA

Figure 3. A quadric degenerating to the union of two planesFigure 4. Chain of n quadrics as in Example 3.2Each quadric Yi meets Y n Yi either along a line or along two distinct lines of the sameruling. Thus, as we noted before, the quadric Yi degenerates, in the P3 spanned by Yi, to theunion of two planes meeting along a line li, leaving the other line(s) �xed. Therefore, in P2n+1,the scroll S degenerates also to a planar Zappatic surface X of degree 2n. The line li can bechosen generally enough so that X has 2n� 2 points of type R3 as Zappatic singularities, foreach i, i.e. its dual graph GX is a chain of length 2n, see Figure 5 (cf. Remark 2.6).� � � � � � � � � � � �Figure 5. Planar Zappatic surface of degree 2n with a chain as dual graphExample 3.3 (Elliptic scrolls). Let C be a smooth, elliptic normal curve of degree n in Pn�1.Since C degenerates to a union of n lines whose dual graph is a cycle, the smooth ellipticnormal scroll S = C � P1 � P2n�1 degenerates to a Zappatic surface Y = Sni=1 Yi, such thateach Yi is quadric, Y has no Zappatic singularity and its dual graph GY is a cycle of lengthn, see the picture on the left in Figure 6.

�������� � � � �Figure 6. Cycle of n quadrics and of 2n planes as in Example 3.3Each quadric Yi meets Y n Yi along two distinct lines ri; r0i of the same ruling. Hence, inthe P3 spanned by Yi, the quadric Yi degenerates to the union of two planes meeting along aline li, leaving ri; r0i �xed. Choosing again a general li for each i, it follows that in P2n�1 the

DEGENERATIONS OF SCROLLS TO UNIONS OF PLANES 7scroll S degenerates to a planar Zappatic surface X of degree 2n with 2n points of type R3as Zappatic singularities and its dual graph GX is a cycle of length 2n, see Figure 6.Example 3.4 (Abelian surfaces). Let C � Pn�1 and C 0 � Pm�1 be smooth, elliptic normalcurves of degree respectively n and m. Then C and C 0 degenerate to the stick curves CEn andCEm respectively, hence the abelian surface S = C � C 0 � Pnm�1 degenerates to a Zappaticsurface which is a union of mn quadrics with only E4-points as Zappatic singularities, cf. e.g.the picture on the left in Figure 7, where the top edges have to be identi�ed with the bottomones, similarly the left edges have to be identi�ed with the right ones. Thus the top quadricsmeet the bottom quadrics and the quadrics on the left meet the quadrics on the right.

Figure 7. nm quadrics with E4-points and 2nm planes with E6-pointsAgain each quadric degenerates to the union of two planes. By doing this as depictedin Figure 7, one gets a pillow degeneration of a general abelian surface with a polarizationof type (n;m) to a planar Zappatic surface of degree 2nm with only E6-points as Zappaticsingularities.Other examples of pillow degenerations to union of planes are also considered in e.g. [8].Remark 3.5. Going back to the general case, if either C or C 0 has genus greater than 1 andif they degenerate to stick curves, then the surface S = C � C 0 degenerates to a union ofquadrics, as we said. Unfortunately it is not clear if it is possible to further independentlydegenerate each quadric to two planes.From now on, until the end of this section, we deal with degenerations of rational normalscrolls only. Namely we will show that a general rational normal scroll degenerates to a planarZappatic surface with Zappatic singularities of type R3 only and we will see how \general"the scroll has to be in order to admit such degenerations (e.g., in Example 3.2, the scrolls areactually forced to have even degree).There are several ways to construct these degenerations. We will start from the trivialfamily and then we will perform two basic operations: (1) blowing-ups and blowing-downs inthe central �bre, (2) twisting the hyperplane bundle by a component of the central �bre.Construction 3.6. Let S = Sa;b be a smooth, rational, normal scroll of type (a; b) in Pr,where r = a + b + 1 � 3 and we assume that b � a. Then S degenerates to the union of aplane and a smooth, rational normal scroll Sa;b�1 meeting the plane along a ruling.Indeed, S is the embedding of the Hirzebruch surface Fn , n = b�a � 0, via the linear systemjC + aF j, where F is the ruling and C is a section of self-intersection n (clearly, if n = 0,we may choose F to be either one of the two rulings and C to be the other ruling). SetH = C + aF . Consider the trivial family S = Fn � � ��! �. On S we have the hyperplanebundle H which coincides with H on each �bre of �.

8 A. CALABRI, C. CILIBERTO, F. FLAMINI, R. MIRANDANow blow up S at a general point of the central �bre S0. Let V be the exceptional divisorand S 0 be the proper transform of S0. Then, HO(�V ) embeds V as a plane and maps S 0to a scroll of type (a; b � 1), which meet each other along a ruling of S 0. We explain theseoperations in Figure 8, where the dotted lines represent the hyperplane bundle. The last arrowis the so-called type I transformation on the vertical (�1)-curve (cf. [11]), which consists inblowing up the (�1)-curve and then blowing down the exceptional divisor, which is a F0 ,along the other ruling. The total e�ect on S0 is to perform an elementary transformation.When r = 3 this process gives the degeneration of a smooth quadric to two planes meetingalong a line.�C pnF 0a �n 0 blow-up p�����! n�10a �1 �1�n 1 1 1 twist by O(�V )���������! n�10a �1 �1�n 11 1 ! n�10a �n+1 0 10 �1 0

Figure 8. Degeneration of a scroll Sa;b to the union of a plane and a scroll Sa;b�1Construction 3.7. Let S = Sa;b be a smooth, rational, normal scroll of type (a; b) in Pr,where r = a+ b+1 and assume that b � a > 1. Then S degenerates to the union of a quadricand a smooth, rational normal scroll Sa�1;b�1 meeting the quadric along a ruling.Indeed, consider the Hirzebruch surface Fn , n = b � a � 0, and the trivial family S =Fn �� ��! �, with the hyperplane bundle H, as in Construction 3.6.Now blow up a ruling F0 in the central �bre S0. Let W be the exceptional divisor and S 0be the proper transform of S0. Then HO(�W ) embeds W as a quadric and S 0 as a scrollof type (a� 1; b� 1), which meet along a ruling of S 0, cf. Figure 9.CnF 0a �n 0 F0 blow-up F0������! n0a �n 0 00 0 0 twist by O(�W )���������! n0a�1 �n 0 00 0 0

Figure 9. Degeneration of a scroll Sa;b to the union of a quadric and a scroll Sa�1;b�1By induction on the degree of the scroll and by using Constructions 3.6 and 3.7 for theinductive steps, we now show the following:Proposition 3.8. Let d � 2 and set r = d + 1 � 3. Let X := Xd;0 be a planar Zappaticsurface of degree d in Pr, whose dual graph is a chain, i.e. X has d� 2 points of type R3 asZappatic singularities. Then, the Hilbert point of X belongs to the irreducible component Hd;0of the Hilbert scheme parametrizing rational normal scrolls of degree d.Remark 3.9. It is well-known (cf. e.g. Lemma 3 in [6]) that Hd;0 is generically reduced andof dimension d2 + 4d� 3.Proof of Proposition 3.8. We will directly show that a smooth, balanced scroll S degeneratesto X.

DEGENERATIONS OF SCROLLS TO UNIONS OF PLANES 9Suppose �rst that r is even. Let S = S(a; a+ 1) be a balanced scroll of degree d in Pr, i.e.a = (d� 1)=2 = r=2� 1. Consider the trivial family F1 ��, where F1 is embedded in Pr bythe linear system jC + aF j, such as in Constructions 3.6 and 3.7, cf. the picture on the leftin Figure 10.Now blow up a ruling in the central �bre, call W �= F0 the exceptional divisor and twistthe hyperplane bundle by O(�aW ). In this way, one gets a degeneration of S to the unionof a scroll of type (a; a) in Pr�1 and a plane, meeting along a ruling, cf. Construction 3.7 andthe picture in the middle of Figure 10.Then blow up a general point (the bottom left corner in Figure 10) of the scroll, twistagain by the opposite of the new surface and perform a type I transformation, as we didin Construction 3.6. By twisting again by the opposite of the new surface, counted withmultiplicity a� 1, one gets the con�guration depicted on the right in Figure 10, namely the�rst two components are two planes, whereas the new component is a scroll of type (a�1; a).Going on by induction on a, by following the same process, one gets a chain of planes whichis a planar Zappatic surface with only R3-points, as wanted.If r is odd, one starts from a F0 as in the central picture of Figure 10 and one may performexactly the same operations in order to get a similar degeneration. �Remark 3.10. In practice, Proposition 3.8 follows by Contructions 3.6 and 3.7 with a suit-able induction. The explicit argument we made in the proof shows that there exists a atdegeneration of smooth, rational scrolls to X whose total space is singular only at the R3-points of X. For another approach, the reader is also referred to [20].10a �1 0 ! � 00a 0 0 10 �1 0 ! 10a�1 �1 0 �10 1 0 10 �1 0Figure 10. Degeneration of Sa;a+1 to a planar Zappatic surface with only R3-pointsRemark 3.11. Suppose to have a smooth scroll S which is the general �bre of an embeddeddegeneration in Pr to a Zappatic planar surface X. The ruling of S, considered as a curve �in the Grasmannian G (1; r), accordingly degenerates to a stick-curve �0. This means that theruling degenerates to a union of pencils of lines, one in each plane of X. Since �0 is connected,each double line ofX belongs to the pencil in either one of the two planes containing it. Hence,the centers of the pencils also belong to the double lines of X. Therefore, on each plane whichcontains more than one double line of X, all the double lines pass through the same Zappaticsingularity which is the center of the pencil. However, the location of the centers of thepencils on the planes containing only one double line of X is not predictable.We conclude this section by proving the following:Proposition 3.12. Let S = Sa;b be a smooth, rational normal scroll in Pa+b+1, with b�a � 4.Assume that S is the general �bre of a degeneration whose central �bre is a planar Zappaticsurface X. Then X has worse singularities than R3-points.Proof. By construction of the scroll S (cf. De�nition 2.15), the minimum degree of a sectionof S is a and let Ca be the section of degree a. Suppose by contradiction that S is the general�bre of an embedded degeneration of surfaces whose central �bre is a planar Zappatic surface

10 A. CALABRI, C. CILIBERTO, F. FLAMINI, R. MIRANDAX = Sa+bi=1 Vi in Pa+b+1, with only R3-points as Zappatic singularities. Then the dual graphGX is a chain and we may and will assume that two planes Vi and Vj meet along a line if andonly if j = i� 1.While S degenerates to X, the ruling of S degenerates to a pencil of lines �i on each planeVi, i = 1; : : : ; a + b (cf. Remark 3.11) and the section Ca degenerates to a chain of linesl1; : : : ; la, with li � Vji, i = 1; : : : ; a, and we may and will assume that j1 < j2 < � � � < ja.The pencil �1 has to meet Sai=1 li, hence V1 has to have non-empty intersection withVj1, therefore the assumption that X has at most R3-points implies that j1 � 3. For eachk = 2; : : : ; a, the lines lk and lk�1 meet at a point, so the same argument implies thatjk � jk�1 + 2 (cf. Figure 11). It follows that ja � j1 + 2(a� 1) � 2a+ 1.On the other hand, the pencil �a+b has to meet Sai=1 li, hence ja � a+ b�2. In conclusion,one has that: a + b� 2 � ja � 2a+ 1;which contradicts the assumption that b � a+ 4. �l1 l2 l3 : : : laFigure 11. Degeneration of Sa;b, b = a+ 3, to X with only R3-pointsFor another approach, the reader is referred to [20].Remark 3.13. By following the lines of the proof of Proposition 3.8 it is possible to provethat, given a; b positive integers such that 0 � b � a � 3, there exist degenerations whosegeneral �bre is a scroll of type S(a; b) and whose central �bre is a planar Zappatic surfacewith only R3-points as Zappatic singularities (cf. Figure 11). We will not dwell on this here.4. Degenerations of scrolls: inductive constructionsIn this section we produce families of smooth scrolls of any genus g � 0 which degenerateto planar Zappatic surfaces with Zappatic singularities of types R3 and S4 only.We start by describing the planar Zappatic surfaces which will be the limits of our scrolls.We will construct these Zappatic surfaces by induction on g. From now on in this section,we will denote by Xd;g a planar Zappatic surface consisting of d planes and whose sectionalgenus is g.We start with the case g = 1.Construction 4.1. For any d � 5, there exists a planar Zappatic surface Xd;1 = Sdi=1 Vi inPr, with r = d� 1, whose dual graph is a cycle.Indeed, if p1; : : : ; pd are the coordinate points of Pr, we may let Vi, i = 2; : : : ; d � 1,be the plane spanned by pi�1; pi; pi+1 and let V1 = hpd; p1; p2i, Vd = hpd�1; pd; p1i. ThenXd;1 = Sdi=1 Vi is a planar Zappatic surface with dual graph a cycle and whose Zappaticsingularities are points of type R3 at p1; : : : ; pd, cf. Figure 12, where one identi�es the linehpd; p1i on the left with the same line on the right.

DEGENERATIONS OF SCROLLS TO UNIONS OF PLANES 11� ��� ���������pd p1 p2 p3 p4 p5 p6 pd�4 pd�3 pd�2 pd�1 pd p1Figure 12. Planar Zappatic surface Xd;1 with dual graph a cycleWe will show in Theorem 4.6 that Xd;1 is the at limit of a smooth scroll of genus 1 in Pr.In order to do that, now we describe another way to construct Xd;1, which will also help tounderstand the next inductive steps.Let Xd�2;0 = Sd�2i=1 Vi be a planar Zappatic surface of degree d� 2 in Pr, whose dual graphis a chain. We may and will assume that the planes Vi and Vj meet along a line if and onlyif j = i� 1.Now choose a general line l1 in V1 and a general line l2 in Vd�2, thus l1 [resp. l2] does notpass through the R3-point V1 \ V2 \ V3 [resp. Vd�4 \ Vd�3 \ Vd�2]. Clearly the lines l1 andl2 are skew and span a P3, call it �. By a computation in coordinates one proves that, ifd � 6, then � \X0 = l1 [ l2. Therefore there exists a smooth quadric Q0 in � such that l1,l2 are lines of the same ruling on Q0 and Q0 meets X0 transversally along Q0 \X0 = l1 [ l2.On the other hand, if d = 5, then � \X0 = l1 [ l2 [ l, where l is a line in the central plane.Nonetheless it is still true that there exists a smooth quadric Q0 which contains l1 and l2 andmeets X0 transversally.Finally, in �, the quadric Q0 degenerates to two planes Vd�1 and Vd, such that li � Vd�i+1,i = 1; 2. By construction, the planar Zappatic surface Xd;1 = Xd�2;0 [ Vd�1 [ Vd = Sdi=1 Vihas dual graph which is a cycle, hence it has only R3-points as Zappatic singularities (cf.Example 3.3 and Figure 13). Note that, if d � 6, then there are pairs of disjoint planes inthe cycle.

l1 l2V1 Vd�2Q0 l1 l2V1 Vd�2Vd Vd�1Figure 13. Construction of Xd;1 from Xd�2;0Next, we complete the construction proceeding inductively.Construction 4.2. Fix integers g; d such that g � 2 and d � 2g+ 4. Set c = d� 2g� 4 � 0and r = 5 + c = d � 2g + 1. There is a planar Zappatic surface Xd;g = Sdi=1 Vi in Pr suchthat: � Xd;g has 3g + 6 + c double lines, i.e. its dual graph GXd;g has 3g + 6 + c edges;� Xd;g has r + 1 points of type R3 and 2(g � 1) points of type S4;� for each i, Vi is the central plane through a point p of type either R3 or S4, i.e. Vi isthe central component of Xd;g passing through p as de�ned in Notation 2.2;

12 A. CALABRI, C. CILIBERTO, F. FLAMINI, R. MIRANDA� there exist two R3-points of Xd;g whose central planes do not meet;� �(OXd;g) = 1� g, p!(Xd;g) = 0, q(Xd;g) = g(Xd;g) = g.Taking into account Construction 4.1, which covers g = 1 and d � 5, we can proceed byinduction and assume that we have the surface Xd�2;g�1. Let V1 and V2 be disjoint planesin Xd�2;g�1 such that each one of them is the central plane for a R3-point, say p1 and p2respectively.Now choose a line l1 in V1 [resp. l2 in V2] which is general among those passing throughp1 [resp. through p2]. Then l1 and l2 are skew and span a P3, say �, therefore there exists asmooth quadric Q0 in � containing l1 and l2 as lines of the same ruling, cf. Figure 14.Now we prove the following:Claim 4.3. For general choices, Q0 and Xd�2;g�1 meet transversally along Xd�2;g�1 \ Q0 =l1 [ l2.Proof. In order to prove the claim, it suÆces to show that � does not meet the remainingcomponents of Xg�1 along a curve, i.e. that � does not meet Vi, i 6= 1; 2, along a line. Beforeproving the claim, we make a remark. Suppose that there are two further planes, say V3 andV4, in Xd�2;g�1 contained in hV1; V2i = � �= P5. Suppose also that the dual graph of theplanar Zappatic surface V1 [ V3 [ V4 [ V2 is a chain of length 4. Then the points V1 \ V3 \ V4and V3 \ V4 \ V2 are of type R3. Note that this certainly happens if c = 0 and g = 2 becausein that case the dual graph of Xd�2;g�1 is a cycle of length six.In this situation, a computation in coordinates in � shows that for a general choice of l1and l2, � = hl1; l2i does not intersect either V3 or V4 along a line.Now we prove the claim arguing by contradiction. Fix the line l2 in V2 and considerhl2; V1i = �= P4. By moving l1 in the pencil of lines of V1 through p1, one gets a pencil �of P3's inside and each of these P3's meets a plane, say V3, along a line. There are twopossibilities: either V3 � , or V3 * .In the former case, V3 intersects V1 at a point q. Let l2 move in the pencil of lines of V2through p2: one gets a pencil of P4's in � = hV1; V2i, whose base-locus is hp1; V2i �= P3 inwhich V3 is contained. This implies that q = p1, moreover V3 intersects V2 along a line whichnecessarily contains p2. In conclusion, V3 contains the line passing through p1 and p2. Thisyields the existence of a plane V4 which forms, together with V1, V2 and V3, a con�gurationin � of four planes as the one discussed above. This is a contradiction.Suppose now that V3 * . Then V3 meets along a line the base locus of the pencil �, whichis the plane hp1; l2i. By moving l2, we see that V3 has to contain the line through p1 and p2and we get a contradiction as before. �In �, the smooth quadric Q0 degenerates to the union of two planes, say Vd�1 [ Vd, whereli � Vd�i+1, i = 1; 2. Consider the planar Zappatic surface Xd;g = Xd�2;g�1 [ Vd�1 [ Vd ofdegree d in Pr. Thus, we added to Xd�2;g�1 two planes and three double lines V1\Vd, Vd\Vd�1and Vd�1 \V2. Moreover, the points p1 and p2 become points of type S4 for Xg and we addedtwo further points of type R3 at V1 \ Vd \ Vd�1 and Vd \ Vd�1 \ V2, cf. Figure 14. Finally,one checks that each one of the planes Vd�1 and Vd is disjoint from some other plane in thecon�guration. This ends the construction.Next, we will prove that the Zappatic surfaces Xd;g we constructed are limits of smoothscrolls of genus g. First we make a remark.

DEGENERATIONS OF SCROLLS TO UNIONS OF PLANES 13�

�p1

p2l1 l2V1 V2Q0 �

�p1

p2l1 l2VdV1 V2Vd�1

Figure 14. Construction of Xd;g from Xd�2;g�1Remark 4.4. If Xd;g is the at limit of a family of smooth surfaces Y , then Theorem 2.12implies that:g(Y ) = g; pg(Y ) = 0; �(OY ) = 1� g; 8(1� g) � K2Y � 6(1� g): (4.5)Theorem 4.6. Let g � 0 and d � 2g + 4 be integers. Let r = d� 2g + 1. The Hilbert pointcorresponding to the planar Zappatic surface Xd;g belongs to an irreducible component Hd;gof the Hilbert scheme of scrolls of degree d and genus g in Pr, such that:(i) the general point of Hd;g represents a smooth, linearly normal scroll Y � Pr;(ii) Hd;g is generically reduced, dim(Hd;g) = h0(Y;NY=Pr) = (r + 1)2 + 7(g � 1), andmoreover h1(Y;NY=Pr) = h2(Y;NY=Pr) = 0.Proof of Theorem 4.6: beginning. We prove Theorem 4.6 by induction on g. The case g = 0has been treated in Proposition 3.8. By induction on g, we may assume that Xd�2;g�1 is the at limit of a smooth scroll S of degree d� 2 and genus g� 1 in Pr, which is represented by asmooth point of a component Hd�2;g�1 of the Hilbert scheme of dimension (r+1)2+7(g�2).We can now choose l1 and l2 as in Constructions 4.1 and 4.2 so that they are limits ofrulings F1 and F2, respectively, on S (cf. Remark 3.11).Let Q be a smooth quadric containing F1 and F2, whose limit is Q0. By the propertiesof Xd�2;g�1 and of Q0 (see Claim 4.3), it follows that S and Q meet transversally alongS \Q = F1 [ F2.The inductive step is a consequence of the following lemma. �Lemma 4.7. In the above setting, consider the unionR := S [Q:Let NR and TR be the normal and the tangent sheaf of R in Pr, respectively; then, one has:H1(NR) = H2(NR) = 0; (4.8)h0(NR) = (r + 1)2 + 7(g � 1) = d2 � 4dg + 4d+ 4g2 � g � 3: (4.9)Furthermore the natural map H0(NR)! H0(T 1), induced by the exact sequence0! TR ! TPrjR ��! NR ! T 1 := Coker(�)! 0; (4.10)is surjective.Proof. We will compute the cohomology of NR, by using a similar technique as in section 2.2of [6] (see Lemma 3 therein).

14 A. CALABRI, C. CILIBERTO, F. FLAMINI, R. MIRANDALet � := S \ Q = F1 [ F2 be the double curve of R. Since R has global normal crossings,the sheaf T 1 in (4.10) is locally free, of rank 1 on the singular locus � of R and, by [10], it isT 1 �= N�=S N�=Q:Since � is the union of two lines of the same ruling on both Q and S, it follows thatT 1 �= O�: (4.11)Let us consider the inclusions �S : NS ! NRjS and �Q : NQ ! NRjQ. Lemma 2 in [6] showsthat T 1 �= coker(�S) and T 1 �= coker(�Q). For readers' convenience, we recall here the proof.By a local computation, one sees that the cokernel K of �S is locally free of rank 1 on �. Inthe diagram TPrjR NR T 1 0NRjSTPrjS NS �S 0 (4.12)the horizontal and diagonal rows are exact, hence the commutativity of the pentagon showsthat T 1 surjects onto K. Since both are locally free sheaves of rank 1, one concludes thatT 1 �= K. The same argument works for Q.Hence the following sequences are exact:0!NS ! NRjS ! T 1 ! 0; (4.13)0!NQ(��)! NRjQ(��)! T 1(��)! 0: (4.14)Moreover, one has the exact sequence0!NRjQ OR(��)! NR !NRjS ! 0; (4.15)so that, in order to prove (4.8), it suÆces to show thatH i(NRjS) = 0; for 1 � i � 2; (4.16)H i(NRjQ OR(��)) = 0; for 1 � i � 2: (4.17)By induction on g, one knows that H i(NS) = 0, i = 1; 2. By (4.11), one has that H i(T 1) =H i(O�) = 0, i = 1; 2, because � is the union of two distinct lines. Hence the sequence (4.13)implies (4.16).Note that H i(T 1(��)) = H i(O�(��)) = 0, i = 1; 2. Taking into account the exactsequence (4.14), the proof of (4.17) is concluded if one shows thatH i(NQ(��)) = 0; for 1 � i � 2: (4.18)Since Q lies in a P3, one has thatNQ �= OQ(2)�OQ(1)�(r�3):Recall that F1 and F2 are lines of the same ruling, so F1 � F2 and OQ(��) �= OQ(�2F1). LetG be the other ruling of Q and H be the general hyperplane section of Q, hence H � G+F1and one has that: NQ OQ(��) �= OQ(2G)�OQ(G� F1)�(r�3) (4.19)and one sees that hi(OQ(2G)) = hi(OQ(G� F )) = 0, for i = 1; 2, which proves (4.18). Theproof of (4.8) is thus concluded.

DEGENERATIONS OF SCROLLS TO UNIONS OF PLANES 15We now prove formula (4.9). By (4.8), one has that h0(NR) = �(NR), which one computesby using (4.13), (4.14) and (4.15):�(NR) = �(NRjS) + �(NRjQ OQ(��)) = �(NS) + �(T 1) + �(NQ(��)) + �(T 1(��)):By (4.11), one has that �(T 1) = �(T 1(��)) = 2. By (4.19), one has �(NQ(��)) = 3. Finally,by induction �(NS) = (r + 1)2 + 7(g � 2);which concludes the proof of (4.9).It remains to show that the map H0(NR)! H0(T 1) is surjective. Since H1(NS) = 0, themap H0(NRjS)! H0(T 1) is surjective by (4.13). Finally (4.17) implies that H0(NR) surjectsonto H0(NRjS), which concludes the proof of the lemma. �We are �nally ready for theProof of Theorem 4.6: conclusion. By Lemma 4.7, one has that H1(NR) = 0, which meansthat R corresponds to a smooth point [R] of the Hilbert scheme of surfaces with degree dand sectional genus g in Pd�2g+1. Therefore, [R] belongs to a single reduced component Hd;gof the Hilbert scheme of dimension h0(NR). The last assertion of Lemma 4.7 implies that ageneral tangent vector to Hd;g at the point [R] represents a �rst-order embedded deformationof R which smooths the double curve �. Therefore, the general point in Hd;g represents asmooth, irreducible surface Y . Thus Y degenerates to R and also to the planar Zappaticsurface Xd;g (cf. Proposition 3.8 and Constructions 4.1, 4.2).Classical adjunction theory (cf. e.g. [17] and x 7 in [9]) implies that Y is a scroll: otherwise,if H is the hyperplane section of Y , one has KY +H nef and therefore 0 < d � 4(g� 1)+K2Ycontradicting K2Y � 6(1� g) in (4.5).Finally, the assertion about linear normality is trivial for g = 0 and is clear by inductionand construction, for g > 0. �Remark 4.20. By using the same �rst part of the proof of Theorem 4.6, one can observethat Construction 4.2 can be carried on also when d = 2g + 3.Indeed, in this case, Xd;g is a union of planes lying in P4 which is not a Zappatic surface ifg � 2, since there are singular points where only two planes of the con�guration meet, whichare not Zappatic singularities. The only di�erence in the construction is that, since there areno pairs of disjoint planes, we have to choose l1 and l2 on two planes V1 and V2 which meetat a point but not along a line. Moreover the proof of the existence of the quadric meetingtransversally the union of planes along l1 [ l2 is a bit more involved.Nonetheless, as in the proof of Theorem 4.6, one can show that Xd;g is a at limit of afamily of linearly normal scrolls in P4 for any genus g � 0 and degree d = 2g + 3. Thesescrolls are smooth only if g = 0; 1, whereas they have isolated double points if g � 2.We �nish this section by mentioning two more examples of con�gurations of planes forminga planar Zappatic surface, with only points of type R3 and S4, which are degenerations ofsmooth scrolls. The advantage of this construction is that they are slightly simpler thanConstruction 4.2. The disadvantage is that they work only for larger values of the degree.Example 4.21. Fix arbitrary integers g; d such that g � 2 and d > 4g. Set r = d� 2g + 1.Let Xd�2g;0 = Sd�2gi=1 Vi be a planar Zappatic surface in Pr whose dual graph is a chain. Onecan attach 2g planes to Xd�2g;0 in order to get a planar Zappatic surface Yd;g of degree d andsectional genus g in Pr with d� 2g + 2 points of type R3 and 2g � 2 points of type S4.

16 A. CALABRI, C. CILIBERTO, F. FLAMINI, R. MIRANDAIndeed, we may assume that Vi meets Vj along a line if and only if j = i � 1. Denote byp2; : : : ; pd�2g�1 the points of type R3 ofXd�2g;0, where pi = Vi�1\Vi\Vi+1, i = 2; : : : ; d�2g�1.Choose a general line l1;1 in V1 [resp. l1;2 in Vd�2g], i.e. a line not passing through p2 [resp.pd�2g�1]. For i = 2; : : : ; g, choose a line li;1 in Vi [resp. a line li;2 in Vd�2g+1�i], which is generalamong those lines passing through pi [resp. through pd�2g+1�i].The generality assumption implies that all the lines li;1; li;2, 1 � i � g, are pairwise skew.For every i = 1; : : : ; g, there is a smooth quadric surface Q0i which contains li;1 and li;2, in theP3 spanned by them. In this P3 the quadric Q0i degenerates to two distinct planes, say Vi;1and Vi;2, leaving li;1 and li;2 �xed: the plane Vi;1 contains li;1 whereas Vi;2 contains li;2. ThenY = Yd;g := Xd�2g;0 [Sgi=1(Vi;1 [Vi;2) is a planar Zappatic surface in Pr. Note that we addedto the points p2; : : : ; pd�2g�1 new Zappatic singularities at the points:(i) qi;j, with 1 � i � g, 1 � j � 2, where qi;1 = Vi\Vi;1\Vi;2 and qi;2 = Vi;1\Vi;2\Vd�2g+1�i,(ii) p1 = V1 \ V2 \ V1;1 and pd�2g = Vd�2g \ Vd�2g�1 \ V1;2Then Y is a planar Zappatic surface with the following properties:� the dual graph GY has d vertices and d+ g � 1 edges;� Y has 2g � 2 points of type S4, namely p2; : : : ; pg; pd�3g+1; : : : ; pd�2g�1;� Y has d� 2g + 2 points of type R3, namely qi;j, 1 � i � g, 1 � j � 2, p1, pd�2g andpg+1; : : : ; pd�3g.� �(OX) = 1� g, p!(X) = 0, q(X) = g(X) = g,(cf. Figure 15).l1;1V1 l2;1V2 l3;1V3 V4 l3;1V5 l2;1V6 l1;1V7 � � � �� � � �� � � ��V1 V2 V3 V4 V5 V6 V7

V1;1 V1;2V2;1 V2;2V3;1 V3;2Figure 15. Construction of Yd;g from Xd�2g;0 for g = 3 and d = 4g + 1 = 13Recall that Xd�2g;0 is the at limit of a smooth, rational normal scroll S of degree d� 2gin Pd�2g+1. If Fi;j, 1 � i � g, 1 � j � 2, is the ruling of S whose limit is li;j and Qi asmooth quadric containing Fi;1, Fi;2, whose limit is Q0i, then one can show, by using similartechniques as in the proof of Theorem 4.6, that the union of the rational normal scroll S andthe g quadrics Qi is a at limit of a family of smooth, linearly normal scrolls of degree d andgenus g in Pd�2g+1, which is contained in a the same component Hd;g of Theorem 4.6 (cf.Theorem 5.4 and Remark 5.5 below).With a slight modi�cation of the previous construction, one can cover also the case d = 4g.We do not dwell on this here.Example 4.22. Fix integers g; d such that g � 1 and d � 3g+2. By induction on g, we willconstruct a planar Zappatic surface Zd;g = Sdi=1 Vi in Pd�2g+1 such that:� Zd;g has d� 2g + 1 double lines, i.e. GZd;g has d� 2g + 1 edges;� Zd;g has d� 2g + 2 points of type R3 and 2g � 2 points of type S4;� for each i, Vi is the central plane through a point p of type either R3 or S4;� there exist two R3-points of Zd;g whose central planes do not meet, unless g = 1 andd = 5;

DEGENERATIONS OF SCROLLS TO UNIONS OF PLANES 17� �(OZd;g) = 1� g, p!(Zd;g) = 0, q(Zd;g) = g(Zd;g) = g.The base of the induction is the case g = 1. In this case, Zd;1 is the surface Xd;1 consideredin Construction 4.1. Now we assume g > 1 and we describe the inductive step.Consider the surface Zd�3;g�1, which sits in Pd�2g, which we suppose to be embedded as ahyperplane in Pd�2g+1.If g = 2 and d = 8, choose two distinct planes V1 and V2 of Z5;1 = X5;1, which do not meetalong a line. Otherwise, choose two distinct planes V1 and V2 of Zd�3;g�1 which are centralfor two R3-points, say p1 and p2, and which span a P5.Choose a line l1 in V1 [resp. l2 in V2] which is general among those lines passing through p1[resp. through p2]. Consider a general P4 in Pd�2g+1 containing l1 and l2.One can show that, in this P4, there is a smooth, rational normal cubic scroll R0 whichcontains l1 and l2 and such that R0 meets transversally Zd�3;g�1 along R0 \Zd�3;g�1 = l1 [ l2.In this P4, the cubic scroll R0 degenerates to a planar Zappatic surface X3;0, consisting ofthree planes, say Vd�2, Vd�1 and Vd, such that l1 � Vd and l2 � Vd�2.We de�ne Zd;g = Zd�3;g�1 [X3;0. We added three planes and four double lines; the pointsp1 and p2 becomes of type S4 for Zd;g and we added three points of type R3 at V1 \Vd�1 \Vd,at V2 \ Vd�2 \ Vd�1 and at Vd�2 \ Vd�1 \ Vd. It is clear the existence of two R3-points whosecentral planes do not meet.Arguing by induction, one may assume that Zd�3;g�1 is the at limit of a smooth, linearlynormal scroll S of degree d � 3 and genus g � 1 in Pd�2g . If Fi, i = 1; 2, is the ruling of Swhose limit is li and R is a smooth, cubic scroll containing F1, F2 as ruling and whose limitis R0, one can show, by using the same proof of Theorem 4.6, that the union S [R is the atlimit of a family of smooth, linearly normal scrolls of degree d and genus g in Pd�2g+1, whichis contained in the same component Hd;g of Theorem 4.6 (cf. Theorem 5.4 and Remark 5.5).5. Hilbert schemes of scrollsIn this section we prove that Hd;g, as determined in Theorem 4.6, is the unique irreduciblecomponent of the Hilbert scheme of scrolls of degree d and genus g in Pd�2g+1 whose generalpoint parametrizes a smooth, linearly normal scroll (cf. Theorem 5.4). This component Hd;gdominates Mg (cf. Remark 5.6).This, together with Construction 4.2 and Theorem 4.6, proves Theorem 1.2 in the intro-duction.On the other hand, we will also construct families of scrolls Y of degree d and genus g inPr, with r > d � 2g + 1, with h1(Y;OY (1)) 6= 0 (cf. Example 5.11). We will also show thatprojections of such scrolls may �ll up components of the Hilbert scheme, di�erent from Hd;g,which may even dominate Mg (cf. Example 5.12).Let C be a smooth curve of genus g and let F �! C be a geometrically ruled surface onC, i.e. F = P(F), for some rank-two vector bundle F on C. Furthermore, we assume thatF is very ample, i.e. F is embedded in Pr, for some r � 3, via the OF (1) bundle as a scrollof degree d = deg(F). From now on, H will denote the hyperplane section of F . A generalhyperplane section H is isomorphic to C, so that we will set LF the line bundle on C �= Hwhich is the restriction of the hyperplane bundle. We will denote by R a general ruling of F ,and more precisely by Rx the ruling mapping to the point x in C.Let Y := C � P1. If L is a line bundle on C, we will set~L := ��1(L) ��2(OP1(1)); (5.1)where �i denotes the projection on the ith-factor, 1 � i � 2.

18 A. CALABRI, C. CILIBERTO, F. FLAMINI, R. MIRANDAProposition 5.2. Let C be a smooth curve of genus g � 0 and let F := P(F) be a geomet-rically ruled surface on C. Assume that deg(F) = d.Then there is a birational map ' : Y 9 9 KFwhich is the composition of d elementary transformations at distinct points of a set � :=fy1; : : : ; ydg � Y lying on d distinct rulings of Y . Moreover,(i) '�(OF (H)) = ~LF ;(ii) '�(jOF (H)j) = j ~LF I�=Y j;Proof. The argument is similar to the one in [12], Prop. 6.2, and in [18]. Indeed, let �be a general linear subspace of codimension two in Pr which is the base locus of a pencilP �= P1 of hyperplanes. By abusing notation, we will denote by P the corresponding pencilof hyperplane sections of F . More speci�cally, we will denote by Ht the hyperplane sectioncorresponding to the point t 2 P1. Then we denote by Z := fz1; : : : ; zdg = F \ P; note thatZ is formed by distinct points on distinct rulings.The map ' : Y 9 9 K F is de�ned by sending the general point (x; t) 2 Y to the pointRx \Ht 2 F . One veri�es that ' is birational and that the indeterminacy locus on F is Z.In order to describe the map ' on Y , note that each point zi maps to a point xi 2 C anddetermines a unique value ti 2 P1 such that Hti contains the ruling Rxi , 1 � i � d. Theindeterminacy locus of ' on Y is � := fy1; : : : ; ydg, where yi = (xi; ti), 1 � i � d.As shown in [12], ' is the composition of the elementary transformations based at thepoints of �. The rest of the assertion immediately follows. �Let � = fy1; : : : ; ydg � Y be a subset formed by d distinct points. We consider the linebundle on C L� := OC(x1 + : : :+ xd); (5.3)where �1(yi) = xi, 1 � i � d.Theorem 5.4. Let g � 0 and d > 2g + 3 be integers. Then there exists a unique irre-ducible component Hd;g of the Hilbert scheme, parametrizing scrolls of degree d and genusg in Pd�2g+1, whose general point represents a smooth scroll F � Pd�2g+1 which is linearlynormal, i.e. h1(F;OF (1)) = 0.Proof. Let U � Hilbd(Y ) be the open subset formed by all � = fy1; : : : ; ydg � Y containing dpoints lying on d distinct �bres and imposing d independent conditions on j ~L�j, which meansdim(j ~L� I�=Y j) = dim(j ~L�j)� d:Note that, by the Kunneth formula, h0( ~L�) = 2h0(L�) = 2(d � g + 1). Thus, dim(j ~L� I�=Y j) = d� 2g + 1: The linear system j ~L�j determines a rational map' : Y 9 9 KPd�2g+1:By Proposition 5.2, every smooth scroll F of degree d and genus g in Pd�2g+1 is the imageof such a map. Therefore, for general � in U , the map ' is birational onto its image F ,which is a smooth scroll of degree d and genus g whose Hilbert point [F ] belongs to a uniquewell-determined component Hd;g of the Hilbert scheme.Note that by (ii) of Proposition 5.2, h1(OF (1)) = h1( ~L� I�=Y ) = 0; therefore, by theRiemann-Roch Theorem, h0(OF (1)) = d� 2g + 2: �

DEGENERATIONS OF SCROLLS TO UNIONS OF PLANES 19Remark 5.5. Observe that the irreducible component Hd;g determined in Theorem 4.6 co-incides with the one determined in Theorem 5.4. The case d = 2g + 3 can also be coveredwith similar arguments. In that case, we have surfaces in P4 which are no longer smooth, butthey have 2g(g � 1) double points as dictated by the double point formula. Nonetheless, thestatement of Theorem 5.4 still holds by substituting F with its normalization.Remark 5.6. The dimension count for Hd;g which has been done in Thereom 4.6 also stemsfrom the proof of Theorem 5.4, which provides a parametric representation of Hd;g. Indeed,the number of parameters on which the general point ofHd;g depends, is given by the followingcount:� 3g � 3 parameters for the class of the curve C in Mg, plus� 2d parameters for the general point in U , plus� (r + 1)2 � 1 parameters for projective transformations in Pr, where r = d � 2g + 1,minus� 2(r � 1) = 2d� 4g parameters for the choice of a codimension-two subspace � in Pr,minus� 3 parameters for projective isomorphisms of the pencil of hyperplanes through � withP1.This computation shows thatHd;g has general moduli, in the sense that the base of the generalscroll [F ] 2 Hd;g is a general point of Mg.Observe that this can also be viewed as a consequence of Theorem 4.6 and more speci�callyof the fact that h1(OF (1)) = 0 for [F ] a general point of the generically smooth componentHd;g.Indeed, if F � Pr, r = d� 2g+1, is a smooth scroll, from the Euler sequence restricted toF , 0! OF ! H0(OF (1))_ OF (1)! TPrjF ! 0;we get that h1(TPrjF ) = 0. Therefore, from the normal sequence of F in Pr0! TF ! TPrjF !NF=Pr ! 0;we get the surjection H0(NF=Pr)!! H1(TF ):Since F is a P1-bundle over C, from the di�erential of the map F �! C, we get a surjectionH1(TF )!! H1(TC);hence H0(NF=Pr)!! H1(TC):which shows that Hd;g dominates Mg.Next, we consider the problem of the existence of components of the Hilbert schemes ofscrolls of degree d and genus g in Pr, with r > d � 2g + 1. First, it is easy to determine anupper-bound for r. This subject has been deeply studied by C. Segre (cf. [22] and [12]). Forthe following lemma, compare [22], x 14.Lemma 5.7. Let g � 1 be an integer. Let C be a smooth curve of genus g and let F = P(F)be a ruled surface on C and d = deg(F) � 2g + 1. Assume that there exists a smooth curvein jOF (1)j. Then, h0(OF (1)) � d� g + 2:The equality holds if and only if F = OC � L, in which case OF (1) maps F to a cone over aprojectively normal curve of degree d and genus g in Pd�g.

20 A. CALABRI, C. CILIBERTO, F. FLAMINI, R. MIRANDAProof. The bound on h0(OF (1)) follows by the Riemann-Roch Theorem on C. If the equalityholds, then C is linearly normally embedded as a curve of degree d and genus g in Pd�g . Itis well-known that this curve is projectively normal (cf. [5], [19] and [21]). Therefore F ismapped to a surface X which is projectively normal, since its general hyperplane section is(cf. [13], Theorem 4.27).On the other hand, X is a scroll of positive genus. Therefore X cannot be smooth, and ithas some isolated singularities. This forces X to be a cone (cf. Claim 4.4 in [7]). Hence, theassertion follows. �Remark 5.8. Let C be a smooth curve of genus g and let F = P(F) be a ruled surface onC and d = deg(F) � 2g + 1. Thend� 2g + 2 � h0(OF (1)) � d� g + 2; (5.9)where the lower bound is immediately implied by the Riemann-Roch Theorem whereas theupper bound is given by the previous lemma. Equivalently,0 � h1(OF (1)) � g; (5.10)where the upper-bound is realized by the cones and the lower-bound by the general scrolls inthe component Hd;g considered above.Any intermediate value i of h1(OF (1)), 1 � i � g, can be actually realized. An easyconstruction is via decomposable bundles as the following example shows.Example 5.11. Let g � 3 and let d � 4g � 1 be integers. Let i be any integer between 1and g. Let C be a smooth, projective curve of genus g with a line bundle L such that jLj isbase-point-free and h1(L) = i. Let D be a general divisor of degree d� deg(L). Notice that,since deg(L) � 2g� 2 and d � 4g� 1, then deg(D) � 2g+ 1 and the linear series jDj is veryample.Consider F = L�OC(D). If F = P(F) then OF (1) is base-point-free and h1(OF (1)) = i.For large values of i, OF (1) is never very ample. For instance, for i = g� 1, C is forced tobe hyperelliptic and L = g12. Thus, the image of F via jOF (1)j has a double line.Similarly, if i = g� 2, either C is hyperelliptic and L = 2g12, or C is trigonal and L = g13 org = 3 and L = !C. In the former case, the image of F has a double conic; in the second case,the image of S has a triple line. Only in the third case, the image of C via jLj is smooth.The analysis is subtle and we do not dwell here on this.Now we consider the question of whether there are other components, di�erent from Hd;g,of the Hilbert scheme of surfaces in Pd�2g+1 whose general point corresponds to a smoothscroll of degree d and genus g. The answer to this question is aÆrmative; in fact one canconstruct such components even with general moduli. In the next example, we show onepossible construction of a component with general moduli. The reader may easily generateother similar constructions.Example 5.12. Let C be a curve with general moduli of genus g = 4l+ �, where 0 � � � 3.Let L be a very-ample, special line bundle of degree m := 3 + g � l with h0(L) = 4. Notethat such a L varies in a family of dimension � := �(g; 3; m) = �.Let d be an integer with either d � 2g + 10, if � = 0; 1, or d � 2g + 11, if � = 2; 3. Setr = d� 2g + 1.Let N be a general line bundle on C of degree d�m. Note that d�m > g + 7+ l. HenceN is very ample (cf. e.g. [1]) and h0(N) = d�m� g + 1.Set F = L�N and X = P(F). Then R + 1 := h0(OX(1)) = h0(L) + h0(M) = r + 1 + l.

DEGENERATIONS OF SCROLLS TO UNIONS OF PLANES 21Since OX(1) is very ample, X is linearly normal embedded in PR as a smooth scroll ofdegree d and genus g, which can be generically projected to Pr to a smooth scroll X 0 with thesame degree and genus, which belongs a certain component H of the Hilbert scheme. As inthe proof of Theorem 4.6, the general member of H is a scroll of the same degree and genus.The dimension of H can be easily bounded from below by the sum of the following quan-tities:� 3g � 3, which are the parameters on which C depends,� g, which are the parameters on which N depends,� �, which are the parameters on which L depends,� (r + 1)l = dim(G (r; R)), which are the parameters for the projections,� (r + 1)2 � 1 = dim(PGL(r + 1; C)):The hypothesis on d implies that dim(H) � dim(Hd;g), which shows that H is di�erentfrom Hd;g.Remark 5.13. The question of understanding how many components of the Hilbert schemeof scrolls there are, and the corresponding image to the moduli space of curves of genus g,is an intriguing one. The previous example suggests that the a complete answer could berather complicated. It also leaves open the question whether Hd;g is the only component withgeneral moduli for 2g + 4 � d � 2g + 10.6. Comments on Zappa's original approachIn [25], Zappa stated a result about embedded degenerations of scrolls of sectional genusg � 2 to unions of planes. His result, in our terminology, reads as Theorem 1.1 in theintroduction.Zappa's arguments rely on a rather intricate analysis of algebro-geometric and topologicaltype of degenerations of hyperplane sections of the scroll and, accordingly, of the branch curveof a general projection of the scroll to a plane.We have not been able to check all the details of this very clever argument. This is oneof the reason why we preferred to solve the problem in a di�erent way, which is the one weexposed in the previous sections. Our approach has the advantage of proving a result in thestyle of Zappa, but with better hypotheses about the degree of the scrolls.However, the idea which Zappa exploits, of degenerating the branch curve of a generalprojection to a plane, is a classical one which goes back to Enriques, Chisini, etc, and certainlydeserves attention. We hope to come back to these ideas in the future.In reading Zappa's paper [25], our attention has been attracted also by another ingredienthe uses which looks interesting on its own. It gives extendability conditions for a curve on ascroll which is not a cone. We �nish this paper by brie y reporting on this. At the the endof the section we brie y summarize Zappa' s argument for the degenerations of the scroll.Let F � P3 be a scroll, which is not a cone over a plane curve. We do not assume F to besmooth. Equivalently, we can look at F as a curve C in the Grassmannian G (1; 3) of lines inP3, which is isomorphic to the Klein hyperquadric in P5 via the Pl�ucker embedding.Let � be a general plane and let � := F \ �. Consider� : C ! �

22 A. CALABRI, C. CILIBERTO, F. FLAMINI, R. MIRANDAthe normalization map. Then, there is a commutative diagramC �� C � G (1; 3) � P5�� � �where � maps a general point x 2 C to the unique line of F passing through �(x), and �maps each point l 2 C, corresponding to a ruling L of F , to the point L \ �.Zappa proves the following nice lemma:Lemma 6.1. (cf. x1 in [25]) In the above setting:��(O�(1)) �= ��(OC(1)):More speci�cally, � is the projection of C from the plane �� � G (1; 3), �lled up by all linesof �.Proof. The assertion follows from the fact that, if r is a line in �, then ��(r) is the section ofthe tangent hyperplane to G (1; 3) at the point of �� corresponding to r. Such a hyperplanecontains ��, and conversely any hyperplane containing �� is of this type. �Zappa notes that an interesting converse of the previous lemma holds.Proposition 6.2. (cf. x 2 in [25]) An irreducible plane curve � is a section of a scroll F � P3of degree d if and only if � is the projection of a curve C of degree d, lying on a smooth quadricQ � P5, and the center of the projection is a plane contained in Q.Proof. One implication is Lemma 6.1. Let us prove the other implication.Suppose that � is the projection of C � Q � P5 from a plane �� � Q. Since all smoothquadrics in P5 are projectively equivalent, we may assume that Q is the Klein hyperquadric.The assertion follows by reversing the argument of the proof of Lemma 6.1. �Proposition 6.2 can be extended in the following way. Let � be a plane curve of degreed and geometric genus g, such that d � g + 6. Set i = h1(C; ��(�(1))). Then, one has thebirational morphism C j��(O�(1))j�������! �C � Pr; (6.3)where r = d� g + i > 5 and the following linear projection:�C ��! � � P2:Proposition 6.4. (cf. x 3 in [25]) In the above setting, � is a plane section of a scroll F inP3, which is not a cone, if and only if �C lies on a quadric of rank 6 in Pr which contains thecenter of the projection ��.Proof. This is an immediate consequence of Proposition 6.2 and can be left to the reader. �Zappa uses Proposition 6.4 to prove that any plane curve of degree d >> g is the planesection of a scroll F which is not a cone. The next proposition is essentially Zappa's resultin x 7 of [25], with an improvement on the bound on d: Zappa's bound is d � 3g + 2.Lemma 6.5. Let g � 0 and let d � 2g+2 be integers. Let �C be an irreducible, smooth curveof degree d and genus g in Pr, r = d� g. Then there exists a quadric of Pr, of rank at most6, which contains �C and a general Pr�3.

DEGENERATIONS OF SCROLLS TO UNIONS OF PLANES 23Proof. Note that a quadric Q of Pr contains a Pr�3 if and only if Q has rank at most 6.Consider the short exact sequence0! I �C=Pr(2)! OPr(2)! O �C(2)! 0:Since d � 2g + 2, one has h0(O �C(2)) = 2d� g + 1 and �C is projectively normal (cf. [5], [19],[21]). Thus h0(I �C=Pr(2)) = �r + 22 �� (2d� g + 1): (6.6)Let � be a general Pr�3 in Pr. Then, from (6.6), one hash0(I �C[�=Pr(2)) � �r + 22 �� �r � 12 �� (2d� g + 1) = d� 2g � 1 > 0: �We need the following lemma:Lemma 6.7. Let �C � Pr be as in Proposition 6.5 and assume that, if g = 0, d � 3. Let � bea Pr�3. The general quadric in the linear system jI �C[�=Pr(2)j has rank k > 3.Proof. Suppose by contradiction that all quadrics containing �C and � have rank 3. Let usde�ne R3( �C) := fQ 2 P(H0(I �C=Pr(2))) j rank(Q) � 3g:By an easy count of parameters our assumption implies that:dimR3( �C) � 3d� 4g � 7:Next, we will show that this inequality is not possible.In order to do that, we apply results from [24]. Zamora proves in [24], cf. Lemma 1.2, thatthere is a one-to-one correspondence between quadrics Q 2 R3( �C) and pairs (g1a; g1b ) of linearseries on �C, with a � b, such that:(i) a + b = deg �C = d,(ii) jg1a + g1b j = jO �C(1)j,(iii) g1a +Bb = g1b +Ba, where Ba (Bb, resp.) is the base locus of the g1a (g1b , respectively).Let Q be the general member of an irreducible component W of maximal dimension ofR3( �C) and let (g1a; g1b ) be the corresponding pair of linear series on �C.Zamora's result implies that there is a base-point-free linear series g1h on �C such thatg1a = g1h +Ba; g1b = g1h +Bb;so that jO �C(1)j = j2g1h +Ba +Bbj:Note that, once the divisor Ba + Bb has been �xed, the line bundle L corresponding to g1hbelongs to a zero-dimensional set in Pich( �C). Set Æ = deg(Ba +Bb), so that d = 2h+ Æ.Suppose now that L is non-special. Then,3d� 4g � 7 � dim(W ) � Æ + 2(h� g � 1) = d� 2g � 2;which gives a contradiction.Now assume that L is special, so that jLj = grh, with 2r � h. In this case3d� 4g � 7 � dim(W ) � Æ + 2(r � 1) � Æ + h� 2;which leads to a contradiction. �

24 A. CALABRI, C. CILIBERTO, F. FLAMINI, R. MIRANDAAs a consequence of the previous lemma, we have:Theorem 6.8. Let � be an irreducible, plane curve of degree d and geometric genus g � 0.If d � max fg + 5; 2g + 2g, then � is a plane section of a scroll in P3, which is not a cone.Proof. Let �C � Pr be the curve corresponding to � in P2. Then � is the projection of �Cfrom � = Pr�3 disjoint from �C. By Lemma 6.5, there is a quadric Q containing �C [ �. Ifrank(Q) := k is 6, we �nished by Proposition 6.4. By Lemma 6.7, we know that k � 4.If k = 5, then the vertex V of Q is a Pr�5. By projecting from V , Q maps to a smoothquadric Q0 in P4 containing C 0, the projection of �C, and �0, the projection of �; the line �0is skew with respect to C 0. Of course � is the projection of C 0 from �0. Let us embedd P4in P5 as a hyperplane. We can certainly �nd a smooth quadric �Q in P5 containing Q0 andcontaining a plane � intersecting the P4 in �0. The curve � is now the projection of C 0 from�. The assertion follows from Proposition 6.2.If k = 4, then the vertex V of Q is a Pr�4. Suppose �rst that � contains V ; then byprojecting from V to P3, the quadric Q maps to a smooth quadric Q0, containing C 0, theimage of �C, and the point p 2 Q0, the image of �, which does not sit on C 0. The curve � is theprojection of C 0 from p. At this point, we can �nish as in the previous case, by embeddingthe P3 in P5 and �nding a smooth quadric �Q in P5 containing Q0 and the plane � intersectingthe P3 in p.If � does not contain V , it intersects V in W �= Pr�5. By projecting from W to P4 we geta situation similar to the case k = 5. The only di�erence is that Q0 is now singular at a pointp, however �0, the projection of �, does not contain p. So we can conclude exactly as in thecase k = 5. �Remark 6.9. We add a little remark to Theorem 6.8. Let � be a plane curve which is aplane section of a scroll F � P3, which is not a cone. So if one applies Theorem 6.8, the scrollwhich extends � is certainly not developable.As Zappa does in [25], one can get an interesting consequence of Theorem 6.8 by applyingduality. Recall that the class of an irreducible plane curve is the degree of the dual curve.Corollary 6.10. An irreducible, plane curve of class d and geometric genus g, such thatd � max fg + 5; 2g + 2g, is the branch curve of a projection of a scroll in P3 of degree d andgenus g, which is not a cone.Proof. Let D � P2 be an irreducible plane curve of class d. Let � � (P2)� be the dualcurve. By Theorem 6.8, � is the plane section of a scroll � which is not a cone. By standardproperties of duality, D is the branch curve of the projection of F = �� from the pointcorresponding to the plane in which � sits. �The argument of Zappa to prove the degeneration of a scroll to a union of planes runs asfollows. Zappa considers the scroll F whose hyperplane section � is a general member of theSeveri variety Vd;g of plane curves of degree d and geometric genus g. Then he lets � degenerateto a general union of d lines. From a complicated analysis involving the degeneration of �and the degeneration of its dual curve, which is the branch curve of the projection of the dualof the surface on the plane (see Corollary 6.10), Zappa deduces that in this degeneration of�, F degenerates to a union of planes. Moreover, he controls the degeneration of the linearlynormal model of F deducing that it also degenerates to a union of planes with only points oftype R3 and S4.

DEGENERATIONS OF SCROLLS TO UNIONS OF PLANES 25References[1] E. Arbarello, M. Cornalba, Su una proprieta' notevole dei mor�smi di una curva a moduli generaliin uno spazio proiettivo, Rend. Sem. Mat. Univers. Politecn. Torino 38 (1980), no. 2, 87{99.[2] A. Calabri, C. Ciliberto, F. Flamini, R. Miranda, On the geometric genus of reducible surfaces anddegenerations of surfaces to unions of planes, in Collino et al. (eds.), The Fano Conference, 277{312,Univ. Torino, Turin, 2004.[3] A. Calabri, C. Ciliberto, F. Flamini, R. Miranda, On the K2 of degenerations of surfaces and themultiple point formula, to appear on Annals of Mathematics, pp. 43.[4] A. Calabri, C. Ciliberto, F. Flamini, R. Miranda, On the genus of reducible surfaces and degenerationsof surfaces, submitted preprint.[5] G. Castelnuovo, Sui multipli di una serie lineare di gruppi di punti appartenenti ad una curvaalgebrica, Rend. Circ. Mat. Palermo 7 (1893), no. 3, 99{119.[6] C. Ciliberto, A.F. Lopez, R. Miranda, Projective degenerations of K3 surfaces, Gaussian maps, andFano threefolds, Invent. Math. 114 (1993), no. 3, 641{667.[7] C. Ciliberto, A.F. Lopez, R. Miranda, Some remarks on the obstructedness of cones over curves oflow genus, Higher dimensional complex varieties, Proceedings of the Trento conference (1994), DeGruyter, Berlin (1996), 167{182.[8] C. Ciliberto, R. Miranda, M. Teicher, Pillow degenerations of K3 surfaces, in Application of Alge-braic geometry to Coding Theory, Physics and Computation. NATO Science Series II (Mathematics,Physics and Chemistry 36 (2001), 53{64.[9] C. Ciliberto, F. Russo, Varieties with minimal secant degree and linear systems of maximal dimensionon surfaces, to appear in Adv. Math..[10] R. Friedman, Global Smoothings of Varieties with Normal Crossings, Ann. Math. 118 (1983), 75{114.[11] R. Friedman,, D.R. Morrison, (eds.,) The birational geometry of degenerations, Progress in Mathe-matics 29, Birkhauser, Boston, 1982.[12] F. Ghione, Quelques r�esultats de Corrado Segre sur les surfaces r�egl�es, Math. Ann. 255 (1981),77{95.[13] S. Greco, Normal varieties. Notes written with the collaboration of A. Di Sante. Inst. Math. 4,Academic Press, London-New York, 1978.[14] P. GriÆths, J. Harris, Principles of Algebraic Geometry, Wiley Classics Library, New York, 1978.[15] J. Harris, On the Severi problem, Invent. Math. 84 (1986), 445{461.[16] R. Hartshorne, Algebraic Geometry, Springer Verlag, New York, 1977.[17] P. Ionescu, Generalized adjunction and applications, Math. Proc. Camb. Phil. Soc, 99 (1986), 467{472.[18] A. Maruyama, M. Nagata, Note on the structure of a ruled surface, J. reine angew. Math. 239(1969), 68{73.[19] M. Mattuck, Varieties de�ned by quadratic equations, in Questions on Algebraic Varieties - CorsoCIME 1969, Rome (1970), 30{100.[20] R. Miranda, Anacapri lectures on degenerations of surfaces, submitted.[21] D. Mumford, Note on the structure of a ruled surface, J. reine angew. Math. 239 (1969), 68{73.[22] C. Segre, Recherches g�en�erales sur les courbes et les surfaces r�egl�ees alg�ebriques, II, Math. Ann. 34(1889), 1{25.[23] F. Severi, Vorlesungen ueber algebraische Geometrie, Teubner, Leipzig, 1921.[24] A. G. Zamora, On the variety of quadrics of rank four containing a projective curve, Boll. UnioneMat. Ital. Sez. B Artic. Ric. Mat. (8) 2 (1999), no. 2, 453{462.[25] G. Zappa, Caratterizzazione delle curve di diramazione delle rigate e spezzamento di queste in sistemidi piani, Rend. Sem. Mat. Univ. Padova 13 (1942), 41{56.[26] G. Zappa, Sulla degenerazione delle super�cie algebriche in sistemi di piani distinti, con applicazioniallo studio delle rigate, Atti R. Accad. d'Italia, Mem. Cl. Sci. FF., MM. e NN. 13 (2) (1943),989{1021.

26 A. CALABRI, C. CILIBERTO, F. FLAMINI, R. MIRANDAE-mail address : [email protected] address : Dipartimento di Matematica, Universit�a degli Studi di Bologna, Piazza di Porta SanDonato, 5 - 40126 Bologna, ItalyE-mail address : [email protected] address : Dipartimento di Matematica, Universit�a degli Studi di Roma Tor Vergata, Via dellaRicerca Scienti�ca - 00133 Roma, ItalyE-mail address : [email protected] address : Dipartimento di Matematica, Universit�a degli Studi di Roma Tor Vergata, Via dellaRicerca Scienti�ca - 00133 Roma, ItalyE-mail address : [email protected] address : Department of Mathematics, 101Weber Building, Colorado State University, Fort Collins,CO 80523-1874, USA